Factor and simplify the algebraic expression (x + 7)^(1/3) - (x + 7)^(4/3).

Step-by-step solution

1. 1

   Pick the common factor by the smaller exponent. Both terms are powers of the same base $x+7$, so the greatest common factor is that base raised to the smaller exponent, $\tfrac13<\tfrac43$. Factoring out $(x+7)^{1/3}$ leaves whole-number work behind.

2. 2

   Divide each term by the common factor. Using $\frac{u^{m}}{u^{n}}=u^{m-n}$: the first term gives $(x+7)^{1/3-1/3}=1$, and the second gives $(x+7)^{4/3-1/3}=(x+7)^{1}=x+7$. Hence $(x+7)^{1/3}-(x+7)^{4/3}=(x+7)^{1/3}\bigl[1-(x+7)\bigr].$

3. 3

   Simplify the bracket. $1-(x+7)=1-x-7=-x-6=-(x+6).$

4. 4

   Write the tidied result. Moving the sign to the front, $(x+7)^{1/3}-(x+7)^{4/3}=-(x+6)(x+7)^{1/3}.$

5. 5

   Check with a value. At $x=1$ the original is $8^{1/3}-8^{4/3}=2-16=-14$, and the factored form gives $-(7)\cdot 8^{1/3}=-7\cdot 2=-14$, so the two agree.